Among known coding methods for low-bit-rate (for example on the order of between 10 kbit/s to 20 kbit/s) speech and audio signals is adaptive coding for orthogonal transform coefficients, such as discrete Fourier transform (DFT) and modified discrete cosine transform (MDCT). In transform coded excitation (TCX) coding used in Non-Patent Literature 1, for example, the influence of amplitude spectral envelopes is eliminated from a coefficient string X[1], . . . , X[N], which is a frequency-domain representation of an input sound signal, to obtain a sequence (a normalized coefficient string XN[1], . . . , XN[N]), which is then encoded by variable length coding. Here, N in the brackets is a positive integer.
Amplitude spectral envelopes can be calculated as follows.
(Step 1) Linear prediction analysis of an input audio digital signal in the time domain (hereinafter referred to as an input audio signal) is performed in each frame, which is a predetermined time segment, to obtain linear predictive coefficients α1, . . . , αP, where P is a positive integer representing a prediction order. For example, according to a P-order autoregressive process, which is an all-pole model, an input audio signal x(t) at a time point t is expressed by Formula (1) with past values x(t−1), . . . , x(t−P) of the signal itself at the past P time points, a prediction residual e(t) and linear predictive coefficients α1, . . . , αP.x(t)=α1x(t−1)+ . . . +αPx(t−P)+e(t)  (1)
(Step 2) The linear predictive coefficients α1, . . . , αp are quantized to obtain quantized linear predictive coefficients {circumflex over ( )}α1, . . . , {circumflex over ( )}αP. The quantized linear predictive coefficients {circumflex over ( )}α1, . . . , {circumflex over ( )}αP are used to obtain an amplitude spectral envelope sequence W[1], . . . , W[N] of the input audio signal at N points. For example, each value W[n] of the amplitude spectral envelope sequence can be obtained in accordance with Formula (2), where n is an integer, 1≤n≤N, exp(⋅) is an exponential function with a base of Napier's constant, j is an imaginary unit, and σ is an amplitude of prediction residual signal.
                              W          ⁡                      [            n            ]                          =                                                            σ                2                                            2                ⁢                                                                  ⁢                π                                      ⁢                          1                                                                                      1                    +                                                                  ∑                                                  p                          =                          1                                                P                                            ⁢                                                                                                    α                            ^                                                    p                                                ⁢                                                                                                  ⁢                                                  exp                          ⁡                                                      (                                                                                          -                                j                                                            ⁢                                                                                                                          ⁢                              2                              ⁢                                                                                                                          ⁢                              π                              ⁢                                                                                                                          ⁢                              np                              ⁢                                                              /                                                            ⁢                              N                                                        )                                                                                                                                                                2                                                                        (        2        )            
Note that a superscript written to the right-hand side of a symbol without brackets represents exponentiation. Specifically, σ2 represents σ squared. While symbols such as “˜” and “{circumflex over ( )}” used in the description are normally to be written above a character that follows each of the symbols, the symbol is written immediately before the character because of notational constraints. In formulas, these symbols are written in their proper positions, i.e. above characters.